Plan C:
Stay with them but as a condition have them baby proof the dog. You have all valid concerns to worry with two small children so if the dog were about you are right, gates will do little and the carte will be more of a problem then a solution for the dog that is used to be about and besides a kid could put finger through the gate or crate and dog could bite just for fun.
Maybe you can do this:
You can protect the paws by having a dog wearing soft home shoes but the brother would need to have him used to wear so he will not start when you arrive. That will take any possibility of scratching out of the picture. The shoes also have traction so no slipping and crashing into people.
https://www.amazon.com/s?k=dog+shoes&i=pets&ref=nb_sb_noss_1
The second and valid problem is biting that can happen. It is also very easy solution. Any dog can bite a little child in play or out of curiosity if the dog is not used to kids especially and it is very easily preventable not by trusting the dog but by doing something that will eliminate the possibility:
Soft muzzle. Yes, it is not funny at first but dog would rather wear this then to be out of the picture.
It is not that bad, dogs often wear it in public places or during travel so they can not bite out of fear or stress.
Perhaps that two simple solutions will help you to enjoy vacation with the family. They need to do some work, that is for sure but the well being of your kids is a valid concern. Very little discomfort for the dog and a lot of protection for everyone.